We consider the communication complexity of some fundamental convex optimization problems in the point-to-point (coordinator) and blackboard communication models. We strengthen known bounds for approximately solving linear regression, $p$-norm regression (for $1\leq p\leq 2$), linear programming, minimizing the sum of finitely many convex nonsmooth functions with varying supports, and low rank approximation; for a number of these fundamental problems our bounds are nearly optimal, as proven by our lower bounds. Among our techniques, we use the notion of block leverage scores, which have been relatively unexplored in this context, as well as dropping all but the ``middle" bits in Richardson-style algorithms. We also introduce a new communication problem for accurately approximating inner products and establish a lower bound using the spherical Radon transform. Our lower bound can be used to show the first separation of linear programming and linear systems in the distributed model when the number of constraints is polynomial, addressing an open question in prior work.

翻译：我们考虑了点到点（协调器）和黑板通信模型下一些基本凸优化问题的通信复杂度。我们加强了对若干问题的已知界限，包括近似求解线性回归、$p$-范数回归（$1\leq p\leq 2$）、线性规划、最小化具有不同支持集的有限多个凸非光滑函数之和以及低秩近似；对于其中许多基本问题，我们的界限几乎是最优的，这已由下界证明。在我们的技术中，我们利用了块杠杆分数（该概念在相关领域尚未被充分探索），以及在理查森型算法中仅保留“中间”比特的方法。我们还引入了一个新的通信问题，用于精确近似内积，并通过球面拉东变换建立了下界。我们的下界可用于证明当约束数量为多项式时分布式模型中线性规划与线性系统之间的首次分离，从而解决了先前工作中的开放问题。